I felt like following up on Kate's question. There were some good motivational answers there.

Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M and N, whose morphism categories are the categories of bimodules, and whose composition is described by some kind of Connes product. If I restrict to the endomorphism category of M, I get a monoidal category structure, but I don't know how to say anything about it. Here's a barrage of questions:

When people talk about fusion categories coming from subfactors, are they referring to the endomorphism category of one of the factors?

How are the endomorphism categories of M and N related? Are they equivalent? Are they Koszul dual?

Does the Jones index say something concrete about the category, like Frobenius-Perron dimension? (How does one compute Jones index, anyway?)

How do people go about constructing exotic subfactors? Do they just arise in nature? I'm totally okay with pointers to references here.

(bonus) I should get a braided tensor structure from a net of factors on a circle. Is this the center of the fusion category, and is it in the literature?

Edit: Based on the (fantastically illuminating) responses, it seems that my bonus question doesn't make sense, because the M-M bimodule fusion category depends on the choice of N in an essential way. Maybe the phrase "conformal defect" should be used somewhere. If I come up with a suitable replacement, I'll present it as a separate question.

6 Answers
6

1- Usually the fusion category is the category of bifinite correspondences, i.e. Hilbert spaces with actions of $N$ and $M$ whose module dimensions are finite. Jones has a result saying that a bifinite correspondence is irreducible if and only if the algebraic module of bounded vectors is irreducible (on his website, two subfactors and the algebraic decomposition...). This means the fusion category of bifinite correspondences should be equivalent to the fusion category of algebraic bimodules (you probably need bifinite in the sense of Lueck, but this is very technical). The former category is generated by the $N-M$ correspondence $L^(M)$, and the later category is generated by $M$ as an $N-M$ bimodule (generated in the sense of taking tensor products and decomposing into irreducibles). In fact, Morrison, Peters, and Snyder use the algebraic category in their recent paper on extended Haagerup (arXiv:0909.4099v1).

2- This isn't what you're asking, but $L^2(M)$ as an $N-M$ bimodule is a Morita equivalence from $N$-Hilbert modules to $M_1$-Hilbert modules where $M_1$ is the basic construction of $N\subset M$. I just think it's an interesting point to bring up.

4- One of the best ways of constructing subfactors is via planar algebras. Given a suitable fusion category, one can construct a planar algebra. Typical examples of these nice fusion categories are the fusion categories arising from the representation theory of a finite group or a quantum group. This gives rise to a family of subfactors. In fact, since (for finite groups) there are only finitely many irreducible representations, we have that this planar algebra will be finite depth (see arXiv:0808.0764, section 4.1), and the subfactor constructed from this planar algebra will be finite depth as well. When someone says "exotic subfactor," they mean a finite index, finite depth subfactor that doesn't appear in the well known families coming from these fusion categories. To date, the best way of constructing these subfactors is to stumble upon a finite bipartite graph which doesn't appear as a fusion graph determine if it can be a principal graph for a subfactor. This has inspired a program to classify all principal graphs which can occur (see the extended Haagerup paper for a synopsis of this as well).

Tie in to 3- Two exotic subfactors, namely the Haagerup and extended Haagerup subfactor, have been constructed by finiding a subfactor planar algebra with the appropriate principal graph inside the graph planar algebra of the bipartite graph (this technique was first explored in detail in Peters' thesis). These subfactors have index equal to the square of the norm of the graph, which is the Perron-Frobenius eigenvalue. In fact, if a finite index subfactor is extremal (irreducible implies extremal), then the norm squared of the principal graph is always the Jones index. (One typically computes Jones index by computing the von Neumann dimension of the $N$-Hilbert module $L^2(M)$.)

5- I know that Kawahigashi et al. (see arXiv:0811.4128) have found a net of type $III_1$-factors corresponding to intervals on the circle. I would recommend starting there.

form a 3-category CN. Given a net N \in CN, the endomorphisms of the identity on N, that is End(1_N), is a braided tensor category, automatically. (Indeed, the most compact definition of a braided tensor category is a 3-category with one object and one 1-morphism.) We might call End(1_N) the representation category of the net, and denote it Rep(N). The category Rep(N) is related to Drinfeld centers, but that's probably a separate discussion.

(By the way, you don't need to know what a conformal defect is to say what Rep(N) is: a representation of a net N is just a Hilbert space with compatible actions of all the vN algebras occuring in the net on the circle.)

(Evan Jenkins took very faithful notes from a talk I gave about this stuff, which you can find here:

Suppose M and N are infinite factors (in the sense that there is no finite trace), then M-N bimodules correspond to morphisms from N to M. Connes' fusion of bimodules corresponds to composition of morphisms.
Nothing really changes if your factors are finite, you just have to consider ampli-morphisms.

This is a 2-category, as you pointed out, and it has (for free) some additional structure which I would describe as a "spherical 2-C* category". In particular each 1-morphism has a dual, a categorical trace and a categorical dimension and so on (technically, this is a consequence of the existence of a Pimsner-Popa basis).

Finite index corresponds to 1-morphisms with finite categorical dimension. More precisely, the minimal (subfactor) index is the square of the categorical dimension.

Suppose you have a morphism \rho N into M (or equivalently, an N-M bimodule). Then you have a dual \bar \rho M into N. These two 1-morphisms generate your 2-category.

Often when N is a subfactor of M and \rho is the inclusion morphism, people tend to overlook domains of morphisms, and talk only about a tensor category of endomorphisms of M. This can be confusing.

In any case, we are always talking about the 2-category generated by \rho and \bar \rho. The whole category of endomorphisms of M is huge, you don't want that.

1) Two possibilities: people are referring to the category of M - M morphisms (equivalently, bimodules) generated by \rho \otimes \bar \rho; people are overlooking the domain of morphisms (as above) viewing \rho and \bar \rho as elements of the tensor category of M endomorphisms and they are talking about the tensor category generated by these two.
By generated I mean complete w.r.t. composition, projections, direct sum... Beware, only alternate compositions of \rho and \bar rho are allowed (so it's really a 2 category).

The answer to 3) is very simple, and hasn't already been given, so I'll just tell you that.

The Jones index of a subfactor N < M is the "coupling constant" or "Murray-von Neumann dimension" of M as a left N-module. That is, every left N-module is a direct sum of the form

M = N + N + ... + N + Np

where p is some projection in N, and the coupling constant is defined to be the number of copies of N appearing, plus that trace of p. So it's a real number at least 1. Jones proved that for a subfactor, the index is in fact in the set {4 \cos^2(\pi/k)}_k \union [4,\infty].

Alternatively, we can look at the principal graphs the for 2-category of {N,M}-bimodules. (Here N and M are the objects in a 2-category, the bimodules between them are the 1-morphisms, and the intertwiners are the 2-morphisms. You can use either 'algebraic' bimodules or 'L^2' bimodules, per Dave Penneys answer.) The graph has a vertex for each simple bimodule, and an edge between V and W for each copy of W that appears in V (x) M. Here you have to interpret M as a either an N-M bimodule or an M-N bimodule, depending on what sort of bimodule V is.

This graph has a 'norm', the largest eigenvalue of its adjacency matrix. This is also known as the Frobenius-Perron dimenion of M in this 2-category. At least for finite-index finite-depth (that is, finitely many simples) subfactors, the square of this norm is the index.

There are lots of great answers above, I'm just going to try for as succinct of answers as I can to all the questions.

In the II_1 case, the category is the subcategory of the category of bimodules over N tensor generated by M. (Similarly you get another category by reversing the roles of N and M). For III_1 factors you can use endomorphisms instead of bimodules. This tensor category is fusion iff the subfactor is finite depth. You can consider either algebraic bimodules or L^2 bimodules and you end up with the same category by a preprint of Vaughan's.

Morita equivalent with the equivalence given by an appropriate bimodule category of N-M bimodules.

The Jones index measures the FP dimension not of the whole category, but of your particular favorite object M.

There are no deeply satisfying constructions yet. All the constructions are by hands and somewhat messy. The descriptions in Emily Peters thesis and related papers at least give a nice description of the category by generators and relations, but the proof that they exist is still very computational. Everyone would love if someone managed to give a really conceptual construction of one of these beasts.

Further on 1), it's important to realise that there really are two fusion categories coming from each subfactor, the N-N bimodules and the M-M bimodules. Often this isn't mentioned, and causes needless confusion. These days we (me and Noah) tend to refer to these as "the even part" and "the dual even part".

There are also some cases where, although the 2 even parts are necessarily Morita equivalent (and equivalently have the same double) they appear quite different -- e.g. one has commutative tensor product, the other doesn't. The exotic 'Haagerup' subfactor is the first I can think of like this.